Let G be a group given in terms of generators and denning relations. The order problem is said to be solvable for (the given presentation of) the group G if, given any element W of G (as a word in the given generators of G), we can determine the order of W in G. The power problem is solvable for G if, given any pair X, Y of elements of G, we can determine whether or not X belongs to the cyclic subgroup {Y} of G generated by Y. It is easy to see that if either of these problems is solvable for G, then the word problem is also solvable for G.

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.

A right [left] unitary S-system is a set M with right [left] operators in a semigroup S with 1, where x1 = x [1x = x] for all x ∈ M. We define a semigroup S with 1 to be completely right [left] injective provided that every right [left] unitary S-system is injective. The main purpose of this paper is to determine a structure for completely right [left] injective semigroups whose idempotents are in the centre.

G. Lallement [4] has shown that the lattice of congruences, Λ(S), on a completely 0-simple semigroup S is semimodular, thus improving G. B. Preston's result [5] that such a lattice satisfies the Jordan-Dedekind chain condition. More recently, J. M. Howie [2] has given a new and more simple proof of Lallement's result using work due to Tamura [9]. The purpose of this note is to extend the semimodularity result to primitive regular semigroups, to establish a theorem relating certain congruence and quotient lattices, and to provide a theorem for congruences on any regular semigroup.

In [1], the natural representation module of the symmetric groups, hereafter called the first natural representation module of the symmetric groups, was analysed. It is the purpose of this paper to analyse the second natural representation module of the symmetric groups.

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.

In [8] and [9] we initiated a study of lattice theory by means of Baer semigroups. Basically, a Baer semigroup is a multiplicative semigroup with 0 in which the left annihilator L(x) of each element x is a principal left ideal generated by an idempotent, while its right annihilator R(x) is a principal right ideal generated by an idempotent. By [8, Lemma 2, p. 86], L(0) has a unique idempotent generator 1 which is effective as a two-sided multiplicative identity for S. For any Baer semigroup S, if we use set inclusion to partially order both ℒ = ℒ (S) = {L(x) | x ∈ S} and ℛ = ℛ (S) = {R(x) | x ∈ S}, we have by [8, Theorem 5, p. 86], that ℒ and ℛ form dual isomorphic lattices with 0 and 1. The Baer semigroup S is said to coordinatize the lattice L in case ℒ(S) is isomorphic to L. In connection with this, it is important to note that by [9, Theorem 2.3, p. 1214], a poset P with 0 and 1 is a lattice if and only if it can be coordinatized by a Baer semigroup.

Let A = (aij) be an n × n complex matrix. The permanent of this matrix is

where the sum is taken over all permutations p of the set {1, …, n}.

In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.

In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.

In this paper we consider mappings induced by matrix multiplication which are defined on lattices of matrices whose coordinates come from a fixed orthomodular lattice L (i.e. a lattice with an orthocomplementation denoted by ′ in which a ≦ b ⇒ a ∨ (a′ ∧ b) = b). will denote the set of all m × n matrices over L with partial order and lattice operations defined coordinatewise. For conformal matrices A and B the (i,j)th coordinate of the matrix product AB is defined to be (AB)ij = Vk(Aik ∧ BkJ). We assume familiarity with the notation and results of [1]. is an orthomodular lattice and the (lattice) centre of is defined as , where we say that A commutes with B and write . In § 1 it is shown that mappings from into characterized by right multiplication X → XP (P ∈ ) are residuated if and only if p ∈ ℘ (). (Similarly for left multiplication.) This result is used to show the existence of residuated pairs. Hence, in § 2 we are able to extend a result of Blyth [3] which relates invertible and cancellable matrices (see Theorem 3 and its corollaries). Finally, for right (left) multiplication mappings, characterizations are given in § 3 for closure operators, quantifiers, range closed mappings, and Sasaki projections.

All sets considered will be finite, and |x| will denote the cardinal number of the set X.

Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given by

Given a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

A vector lattice W is boundedly complete when each subset {aj:j ∊ J} of W which is bounded above by an element of W has a least upper bound in W. The least upper bound of {aj:j ∊ J} is denoted by and the greatest lower bound by whenever these exist.